Question

5. Consider a $100 par value 8% bond with semiannual coupons called at $106.50 on any coupon date starting 4 years after issue for the next 2 years, at $102 starting 6 years after issue for the next 2 years, and maturing at $100 at the end of 8 years. (a) Find the highest price which an investor can pay and still be certain of a yield of 6% convertible semiannually. (b) Assuming the price paid in part (a), compute the nominal yield convertible semiannually the purchaser would earn if the bond was not called.

Already worked but not correct in chegg.

Answer 1

First of all we will identify all the paths in which bond can be called as shown in the table below

Year	i	i	ii	iii	iv	v	vi	vii	viii
0.5	4	4	4	4	4	4	4	4	4
1	4	4	4	4	4	4	4	4	4
1.5	4	4	4	4	4	4	4	4	4
2	4	4	4	4	4	4	4	4	4
2.5	4	4	4	4	4	4	4	4	4
3	4	4	4	4	4	4	4	4	4
3.5	4	4	4	4	4	4	4	4	4
4	110.5	4	4	4	4	4	4	4	4
4.5		110.5	4	4	4	4	4	4	4
5			110.5	4	4	4	4	4	4
5.5				110.5	4	4	4	4	4
6					106	4	4	4	4
6.5						106	4	4	4
7							106	4	4
7.5								106	4
8									104

Then we will calculate the present value of all these cash flows using 6% as the discount rate (as we want to be certain of 6% semi annual yield) present value is shown in the following table

Year	i	i	ii	iii	iv	v	vi	vii	viii
0.5	3.88	3.88	3.88	3.88	3.88	3.88	3.88	3.88	3.88
1	3.77	3.77	3.77	3.77	3.77	3.77	3.77	3.77	3.77
1.5	3.66	3.66	3.66	3.66	3.66	3.66	3.66	3.66	3.66
2	3.55	3.55	3.55	3.55	3.55	3.55	3.55	3.55	3.55
2.5	3.45	3.45	3.45	3.45	3.45	3.45	3.45	3.45	3.45
3	3.35	3.35	3.35	3.35	3.35	3.35	3.35	3.35	3.35
3.5	3.25	3.25	3.25	3.25	3.25	3.25	3.25	3.25	3.25
4	87.23	3.16	3.16	3.16	3.16	3.16	3.16	3.16	3.16
4.5		84.69	3.07	3.07	3.07	3.07	3.07	3.07	3.07
5			82.22	2.98	2.98	2.98	2.98	2.98	2.98
5.5				79.83	2.89	2.89	2.89	2.89	2.89
6					74.35	2.81	2.81	2.81	2.81
6.5						72.18	2.72	2.72	2.72
7							70.08	2.64	2.64
7.5								68.04	2.57
8									64.81
Sum of PV	112.15	112.77	113.37	113.95	111.36	112.00	112.62	113.22	112.56

From the table above it is clear that an investor can at max pay 111.36 and be certain of atleast 6% YTM irrespective of when the bond is called.

part ii

If the bond is not called till marturity the investor would earn a YTM of

YTM =(C + (F-P) / n ) / (F+P) / 2

YTM = (4 + (100-111.36)/16) / (100+111.36) / 2

YTM = 6.2263